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Synthese

, Volume 186, Issue 2, pp 531–575 | Cite as

A geo-logical solution to the lottery paradox, with applications to conditional logic

  • Hanti Lin
  • Kevin T. KellyEmail author
Article

Abstract

We defend a set of acceptance rules that avoids the lottery paradox, that is closed under classical entailment, and that accepts uncertain propositions without ad hoc restrictions. We show that the rules we recommend provide a semantics that validates exactly Adams’ conditional logic and are exactly the rules that preserve a natural, logical structure over probabilistic credal states that we call probalogic. To motivate probalogic, we first expand classical logic to geo-logic, which fills the entire unit cube, and then we project the upper surfaces of the geo-logical cube onto the plane of probabilistic credal states by means of standard, linear perspective, which may be interpreted as an extension of the classical principle of indifference. Finally, we apply the geometrical/logical methods developed in the paper to prove a series of trivialization theorems against question-invariance as a constraint on acceptance rules and against rational monotonicity as an axiom of conditional logic in situations of uncertainty.

Keywords

Lottery paradox Uncertain acceptance Ramsey test Conditional logic Belief revision Framing effects 

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References

  1. Adams E. W. (1975) The logic of conditionals. D. Reidel, DordrechtGoogle Scholar
  2. Alchourròn C. E., Gärdenfors P., Makinson D. (1985) On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic 50: 510–530CrossRefGoogle Scholar
  3. Arló-Costa H., Parikh R. (2005) Conditional probability and defeasible inference. Journal of Philosophical Logic 34: 97–119CrossRefGoogle Scholar
  4. Barwise K. (1969) Infinitary logic and admissible sets. Journal of Symbolic Logic 34: 226–252CrossRefGoogle Scholar
  5. Douven I. (2002) A new solution to the paradoxes of rational acceptability. British Journal for the Philosophy of Science 53: 391–410CrossRefGoogle Scholar
  6. Hajek P. (1998) Metamathematics of fuzzy logic. Kluwer, DordrechtCrossRefGoogle Scholar
  7. Harper W. (1975) Rational belief change, popper functions and counterfactuals. Synthese 30(1–2): 221–262CrossRefGoogle Scholar
  8. Jeffrey, R. C. (1970). Dracula meets Wolfman: Acceptance vs. partial belief. In M. Swain (Ed.), Induction, acceptance, and rational belief. D. Reidel.Google Scholar
  9. Karp C. (1964) Languages with expressions of infinite length. North Holland, DordrechtGoogle Scholar
  10. Kelly K. (2008) Ockham’s razor, truth, and information. In: van Benthem J., Adriaans P. (eds) Handbook of the philosophy of information (pp. 321–360). Elsevier, DordrechtCrossRefGoogle Scholar
  11. Kelly K. (2011) Ockham’s razor, truth, and probability. In: Bandyopadhyay P., Forster M. (eds) Handbook on the philosophy of statistics.. Elsevier, Dordrecht, pp 983–1024CrossRefGoogle Scholar
  12. Kraus S., Lehmann D., Magidor M. (1990) Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44: 167–207CrossRefGoogle Scholar
  13. Kyburg H. (1961) Probability and the logic of rational belief. Wesleyan University Press, MiddletownGoogle Scholar
  14. Lehmann D., Magidor M. (1992) What does a conditional base entails?. Artificial Intelligence 55: 1–60CrossRefGoogle Scholar
  15. Leitgeb, H. (2010). Reducing belief simpliciter to degrees of belief. Presentation of his unpublished results at the opening celebration of the Center for Formal Epistemology at Carnegie Mellon University in the Summer of 2010 (unpublished results).Google Scholar
  16. Levi, I. (1967). Gambling with truth: An essay on induction and the aims of science. New York: Harper & Row (2nd ed., Cambridge, MA: The MIT Press, 1973).Google Scholar
  17. Levi I. (1969) Information and inference. Synthese 19: 369–391Google Scholar
  18. Levi I. (1996) For the sake of the argument: Ramsey test conditionals, inductive inference and non-monotonic reasoning. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  19. Lin, H. (2011). A new theory of acceptance that solves the lottery paradox and provides a simplified probabilistic semantics for Adams’ logic of conditionals. Master’s thesis, Carnegie Mellon University, Pittsburgh, PA.Google Scholar
  20. Makinson, D., & Gärdenfors, P. (1991). Relations between the logic of theory change and nonmonotonic logic. In A. Fuhrmann & M. Morreau (Eds.), The logic of theory change (pp. 183–205). Springer-Verlag Lecture notes in computer science 465. Berlin: Springer.Google Scholar
  21. Novak V., Perfilieva I., Mockor J. (2000) Mathematical principles of fuzzy logic. Kluwer, DordrechtGoogle Scholar
  22. Pearl, J. (1989). Probabilistic semantics for nonmonotonic reasoning: A survey. In Proceedings of the first international conference on principles of knowledge representation and reasoning (KR ’89) (pp. 505–516). (Reprinted in G. Shafer & J. Pearl (Eds.), Readings in uncertain reasoning (pp. 699–710). San Francisco: Morgan Kaufmann).Google Scholar
  23. Pollock J. (1995) Cognitive carpentry. MIT Press, Cambridge, MAGoogle Scholar
  24. Ramsey, F. P. (1929). General propositions and causality. In H. A. Mellor (Ed.), Philosophical papers. Cambridge: Cambridge University Press; 1990.Google Scholar
  25. Ryan S. (1996) The epistemic virtues of consistency. Synthese 109: 121–141CrossRefGoogle Scholar
  26. van Fraassen B. (1995) Fine-grained opinion, probability and the logic of full belief. Journal of Philosophical Logic 24: 349–377CrossRefGoogle Scholar
  27. Zadeh L. (1965) Fuzzy sets. Information and Control 8: 338–353CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA

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